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Digital Audio

1. Samples and time

Digital audio is stored as a sequence of samples. Each sample is just a numeric value with a certain precision. The sample rate (sr) is the number of samples the audio signal contains per second. Generally speaking, when sample rate increases, the quality of the audio increases.

  1. Define a function that takes in a sample index and sample rate and returns the time value in seconds that correspond to the index.

    • A signal has sample rate = 1. The 1st sample is sampled after 0 seconds, the 2nd sample is sampled after 1 second.
    • A signal has sample rate = 200. The 5th sample is sampled after 0.025 seconds.

  2. Create another function that achieves the same but for an 1-D array of sample indices (e.g. [0, 500, 1000, ...]). You can use your sample_to_time function as long as the output is a numpy array.

    • Calling samples_to_time(np.arange(0, 22050, 512), 22050) should produce the following results: 
      [0.         0.02321995 0.04643991 0.06965986 0.09287982 0.11609977
0.13931973 0.16253968 0.18575964 0.20897959 0.23219955 0.2554195
0.27863946 0.30185941 0.32507937 0.34829932 0.37151927 0.39473923
0.41795918 0.44117914 0.46439909 0.48761905 0.510839   0.53405896
0.55727891 0.58049887 0.60371882 0.62693878 0.65015873 0.67337868
0.69659864 0.71981859 0.74303855 0.7662585  0.78947846 0.81269841
0.83591837 0.85913832 0.88235828 0.90557823 0.92879819 0.95201814
0.9752381  0.99845805]

  3. Create a function gen_time(duration, sr) that creates time stamps in seconds for the given duration at the given sample rate. For e.g. gen_time(1, 10) the output should be

    array([0.        , 0.11111111, 0.22222222, 0.33333333, 0.44444444,
    0.55555556, 0.66666667, 0.77777778, 0.88888889, 1.        ])

2. Creating waves

The frequency of a sine wave can be understood as the number of cycles the sine wave achieves in a second.

  1. (*) How many Radians does it take the sine wave to complete a single cycle?

  2. Create a function single_period() that takes in a sample rate and returns values linearly spaced between 0 and the number of radians at which the sine function completes one cycle. single_period(10) should return

    array([0.        , 0.6981317 , 1.3962634 , 2.0943951 , 2.7925268 ,
3.4906585 , 4.1887902 , 4.88692191, 5.58505361, 6.28318531])

  3. (*) Create a function single_sine(). In this function you should use single_period() to generate the x-axis that the sine function takes as input. Return the results of the sine function. In three different plots, plot the sine wave for the sample rates 10, 50 and 100.

    • Use plt.plot(x, y) where x are the correct time stamps in seconds for each sine wave. You can use a combination of samples_to_time() and single_period() to generate the time stamps.

    • Describe what you see in the plot. What is the total duration in seconds of each sine wave. Explain the difference in duration of each sine wave.

Based on what we have seen so far, it would be beneficial to be able to control more than just the sample rate of our sine waves. We also want to control: * The frequency of the sine wave (i.e. how many cycles the sine wave achieves per second) * The duration of the wave (in seconds) Lets tackle these two parameters one by one.

  1. (*) Define a function sine5(sine_hz, sr) that returns a sine wave with a 5 second duration sampled with the given sr and a frequency corresponding to sine_hz. Plot 3 sine waves in the same figure with sine_hz = 1 and

    • sr = 5
    • sr = 10
    • sr = 100 Use your gen_time() to create the x-axis for each plot. Based on the plot, explain the difference between the sine waves.

  2. (*) Define a function sine(sine_hz, sr, duration) that achieves the same as sine5() but for any duration (in seconds). In the same figure, plot three sines with duration = 2, sr=1000 and:

    • sine_hz = 1
    • sine_hz = 5
    • sine_hz = 20 Again, use gen_time() for the x-axis of each plot.

3. Listen

Instead of just looking at these sine waves, lets try to listen to them.

  1. (*) Generate a 5 second sine wave using your sine() function. The sine wave should have a frequency of 500 Hz sampled at 16000 Hz. Use tools.save_audio() to save your waveform to file. Then open it in e.g. VLC or any media player (Lower the volume if you are wearing headphones!). Describe what you hear.

  2. (*) According to this, the human hearing range is commonly given as 20 - 20,000 Hz. Can you hear a 20Hz and a 20,000 Hz 1 second sine wave? Create a low_sine() and a high_sine() function to generate these waveforms sampled at 48,000 Hz.

    • Use the information from this to determine a minimum sample rate for the high_sine() function.